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Setting to unity the scaling factor for the IR LF of the A1763 core, the other scaling factors are 1.40, 1.14, and 1.27, for the LF of?,,?,, and ?,, respectively.
Setting to unity the scaling factor for the IR LF of the A1763 core, the other scaling factors are 1.40, 1.14, and 1.27, for the LF of, and , respectively.
In addition to the density correction, since the different clusters are located at different redshifts, we rescale the best-fit Schechter parameters obtained for these clusters to the redshift of A1763, adopting the evolution relation of?.
In addition to the density correction, since the different clusters are located at different redshifts, we rescale the best-fit Schechter parameters obtained for these clusters to the redshift of A1763, adopting the evolution relation of.
. The result is shown in Fig. 16..
The result is shown in Fig. \ref{f:irlflit}.
We note that to compare the different luminosity functions, we divide the number densities of the A1763 core IR LF by the logarithmic interval we used for the binning, 0.2.
We note that to compare the different luminosity functions, we divide the number densities of the A1763 core IR LF by the logarithmic interval we used for the binning, 0.2.
The IR LF of the A1763 core lies significantly above all other IR LFs at the faint end, while it is consistent with them at the bright end.
The IR LF of the A1763 core lies significantly above all other IR LFs at the faint end, while it is consistent with them at the bright end.
It is most similar to the IR LF established by for the Bullet cluster.
It is most similar to the IR LF established by for the Bullet cluster.
There clearly seems to be a large variance in the cluster IR LFs, even after correcting for evolutionary effects and after rescaling for the different galaxy densities in the cluster areas sampled by the different surveys.
There clearly seems to be a large variance in the cluster IR LFs, even after correcting for evolutionary effects and after rescaling for the different galaxy densities in the cluster areas sampled by the different surveys.
Part of the variance is caused by observational errors, and the IR LF of?,, which appears to lie below that of in Fig. 16,,
Part of the variance is caused by observational errors, and the IR LF of, which appears to lie below that of in Fig. \ref{f:irlflit},
is consistent with it within the uncertainties?).
is consistent with it within the uncertainties.
. As a source of intrinsic variance, one could consider the effect of an increasing fraction of IR-emitting galaxies with clustercentric radius??).
As a source of intrinsic variance, one could consider the effect of an increasing fraction of IR-emitting galaxies with clustercentric radius.
. However, this trend is far too small to account for the variance we see in the IR LFs, given that they were obtained within rather similar limiting effective radii.
However, this trend is far too small to account for the variance we see in the IR LFs, given that they were obtained within rather similar limiting effective radii.
Moreover, among the four LFs displayed in Fig. 16,,
Moreover, among the four LFs displayed in Fig. \ref{f:irlflit},
that of the A1763 core has been determined within the smallest effective radius, and yet it appears to lie above all the others.
that of the A1763 core has been determined within the smallest effective radius, and yet it appears to lie above all the others.
of UV. optical. and/or near-IR broadband colours to model cluster spectral energy distributions (SED) to simultaneously estimate age and extinction assuming some metallicity (e.g. Pasquali.deGrijs.&Gallagher2003:Hancocketal.2003.2007:Smith 2008).
of UV, optical, and/or near-IR broadband colours to model cluster spectral energy distributions (SED) to simultaneously estimate age and extinction assuming some metallicity (e.g. \citealp{pas03,han03,han07,smi08}) ).
There are several parameters that can affect the colour of a cluster other than age.
There are several parameters that can affect the colour of a cluster other than age.
Reddening plays a very important role. as does the chemical composition.
Reddening plays a very important role, as does the chemical composition.
However. in some age ranges and colours. reddening and chemical composition are degenerate with age.
However, in some age ranges and colours, reddening and chemical composition are degenerate with age.
Furthermore. for low mass clusters (<10" ). the observed integrated colours can be affected by stochastic sampling of the initial mass function IMF) (see for example. Cervino&Luridiana2006.2004:Cervino&Valls-Gabaud2003 and references therein).
Furthermore, for low mass clusters $\la10^{5}$ $_{\odot}$ ), the observed integrated colours can be affected by stochastic sampling of the initial mass function (IMF) (see for example, \citealp{cer06,cer04,cer03} and references therein).
For example. the random addition of a small number of high mass stars will affect the integrated colour of a cluster.
For example, the random addition of a small number of high mass stars will affect the integrated colour of a cluster.
This suggests tha the observed integrated colours of low mass clusters may not be good indicators of age when compared to the integrated colours of a stellar population model with a fully sampled IMF.
This suggests that the observed integrated colours of low mass clusters may not be good indicators of age when compared to the integrated colours of a stellar population model with a fully sampled IMF.
It is not clear how well a particular set of colours can predic the age of a cluster.
It is not clear how well a particular set of colours can predict the age of a cluster.
Are some colour sets better suited than others?
Are some colour sets better suited than others?
What uncertainties can be expected because of the choice of colours in the comparison?
What uncertainties can be expected because of the choice of colours in the comparison?
What uncertainties can be expected because of the assumptions made in generating the model SEDs?
What uncertainties can be expected because of the assumptions made in generating the model SEDs?
Several authors have investigated the use of broadband colours as age indicators by comparison of model colours to the colours of synthetic clusters (e.g. GildePaz&Madore2002:Andersetal.2004:deGrijset 2005).
Several authors have investigated the use of broadband colours as age indicators by comparison of model colours to the colours of synthetic clusters (e.g. \citealp{gil02,and04,deg05}) ).
We use an alternative method by comparing model colours to the integrated colours of resolved Galactic open clusters (OCs).
We use an alternative method by comparing model colours to the integrated colours of resolved Galactic open clusters (OCs).
We compare the published integrated colours of well-studied OCs to a set of population synthesis models.
We compare the published integrated colours of well-studied OCs to a set of population synthesis models.
These clusters have published ages previously determined by the turn-off of the zero-age main sequence on the H-R diagram.
These clusters have published ages previously determined by the turn-off of the zero-age main sequence on the H-R diagram.
This work parallels that of Pessevetal.(2008)... who do a similar analysis of Magellanic Cloud clusters using both optical and IR data.
This work parallels that of \citet{pes08}, who do a similar analysis of Magellanic Cloud clusters using both optical and near-IR data.
The present paper is organized as follows.
The present paper is organized as follows.
In $22 we describe our sample of OCs.
In 2 we describe our sample of OCs.
We describe the population synthesis models used in this study in $33 and the data analysis in $44.
We describe the population synthesis models used in this study in 3 and the data analysis in 4.
Predicting the ages of the OCs. predicting the amount of extinction and the effects of metallicity. stochastic sampling effects and cluster dissolution are discussed in 355.
Predicting the ages of the OCs, predicting the amount of extinction and the effects of metallicity, stochastic sampling effects and cluster dissolution are discussed in 5.
Finally. we summarize in $66.
Finally, we summarize in 6.
We started with the set of Galactic open clusters from. the (Web Base Donnéees Amas) database operated at the Institute for Astronomy of the University of Vienna (Mermilliod 1995)...
We started with the set of Galactic open clusters from the (Web Base Donnéees Amas) database operated at the Institute for Astronomy of the University of Vienna \citep{mer95}. .
The WEBDA database contains 379 OCs. with determined integrated |.B). V. Ry and D colours. and colour excesses. as found by several authors (Lataetal.2002: 19655).
The WEBDA database contains 379 OCs, with well-determined integrated $-$ B), $-$ V), $-$ R), and $-$ I) colours, and colour excesses, as found by several authors \citealp{lat02, bat94, pan89, spa85, sag83, gra65}) ).
We then culled the WEBDA sample to include only the sample of well-studied OCs in Paunzen&Netopil(2006)... who established a list of 72 open clusters with the most accurate known parameters to serve as a standard table for testing isochrones and stellar models.
We then culled the WEBDA sample to include only the sample of well-studied OCs in \citet{pau06}, who established a list of 72 open clusters with the most accurate known parameters to serve as a standard table for testing isochrones and stellar models.
The age uncertainties in Paunzen&Netopil(2006) were determined by measuring the standard deviation of all the published ages in the literature for each of the OCs.
The age uncertainties in \citet{pau06} were determined by measuring the standard deviation of all the published ages in the literature for each of the OCs.
Not all of the 4 integrated colours were determined for euch of the OCs in the Paunzen&Netopil(2006) standard. set.
Not all of the 4 integrated colours were determined for each of the OCs in the \citet{pau06} standard set.
To model observations with unknown dust extinction. we reversed the extinction corrections to the published colours using the published values of aand the conversions from tto the other colour excesses.
To model observations with unknown dust extinction, we reversed the extinction corrections to the published colours using the published values of and the conversions from to the other colour excesses.
We used the conversions in Lataetal.(2002). namely ΕΙ Bi<0.72 Vi0.08 7. R06Vi.and DzI.25V3.
We used the conversions in \citet{lat02}, namely $-$ B)=0.72 $+0.05$ $^2$, $-$ R)=0.6, and $-$ I)=1.25.
From the standard set we created sub-samples of OCs for each of the 4 optical colours and 10 different combinations of colours.
From the standard set we created sub-samples of OCs for each of the 4 optical colours and 10 different combinations of colours.
When multiple integrated colour measurements were available. we adopted the most recently determined values.
When multiple integrated colour measurements were available, we adopted the most recently determined values.
Our final data sets include OCs with mean age uncertainties of and ages ranging from 8 Myr to 8.8 Gyr.
Our final data sets include OCs with mean age uncertainties of and ages ranging from 8 Myr to 8.8 Gyr.
The metallicities associated with this sample ranges from [Fe/H] —1/7 solar to ~2.3 solar.
The metallicities associated with this sample ranges from [Fe/H] $\sim$ 1/7 solar to $\sim$ 2.3 solar.
Unfortunately. the WEBDA database does not give the uncertainties on the total colours for individual OCs.
Unfortunately, the WEBDA database does not give the uncertainties on the total colours for individual OCs.
According to Sagar.Joshi.&Sinvhal(1983). the maximum uncertainty in the published integrated colours for the WEBDA data set is -Ε02 mag.
According to \citet{sag83}, the maximum uncertainty in the published integrated colours for the WEBDA data set is $\pm0.2$ mag.
One ofthe sources of uncertainty listed is the error in the reddening.
One of the sources of uncertainty listed is the error in the reddening.
Because we have used the published colour excesses to reverse the extinction corrections. we can neglect the extinction uncertainty in the colours.
Because we have used the published colour excesses to reverse the extinction corrections, we can neglect the extinction uncertainty in the colours.
Removing this from the total uncertainty. assuming it was originally added in quadrature. the maximum uncertainty in colour is 0.[4.
Removing this from the total uncertainty, assuming it was originally added in quadrature, the maximum uncertainty in colour is $\sim$ 0.14.
We assume that all the measured colours have this maximum uncertainty.
We assume that all the measured colours have this maximum uncertainty.
This assumption further allows us to make fair comparisons of both the accuracy and precision afforded by each colour in age estimation.
This assumption further allows us to make fair comparisons of both the accuracy and precision afforded by each colour in age estimation.
We used a set of evolutionary synthesis models from. theStarburst99 (SB99) code (Leithereretal.1999).
We used a set of evolutionary synthesis models from the (SB99) code \citep{lei99}.
.. We used the new v5.1 eode. which includes the Padova asymptotic giant branch (AGB) stellar models (Vázquez&Leitherer2005).
We used the new v5.1 code, which includes the Padova asymptotic giant branch (AGB) stellar models \citep{vaz05}.
. The new version accounts for all stellar phases that contribute to the integrated light of a stellar population. with arbitrary age from extreme UV to NIR.
The new version accounts for all stellar phases that contribute to the integrated light of a stellar population with arbitrary age from extreme UV to NIR.
Strictly speaking. the Geneva tracks are more appropriate for modeling young clusters. less than 10 Myr. when O stars are present.
Strictly speaking, the Geneva tracks are more appropriate for modeling young clusters, less than 10 Myr, when O stars are present.
Most of our sample OCs have ages greater than 100 Mvr so for simplicity we only consider models with the Padova tracks.
Most of our sample OCs have ages greater than 100 Myr so for simplicity we only consider models with the Padova tracks.
Our SB99 model spectral energy distributions (SEDs) were generated assuming a Kroupa initial mass function (IMF) (favors high mass stars) (<roupa2002) with exponents of 1.3 and 2.3 and mass ranges from 0.10.5 M. and 0.500 M. respectively.
Our SB99 model spectral energy distributions (SEDs) were generated assuming a Kroupa initial mass function (IMF) (favors high mass stars) \citep{kro02} with exponents of 1.3 and 2.3 and mass ranges from $0.1-0.5$ $_{\odot}$ and $0.5-100$ $_{\odot}$ respectively.
We have also assumed instantaneous (single burst) star formation. and solar abundances.
We have also assumed instantaneous (single burst) star formation, and solar abundances.
It has been demonstrated that adopting different forms of the IMF has a minor impact on optical colours (MacArthuretal.2004)... so we have not explored various IMFs.
It has been demonstrated that adopting different forms of the IMF has a minor impact on optical colours \citep{mac04}, so we have not explored various IMFs.
We also generated models with abundances less than ) and greater than ) solar.
We also generated models with abundances less than $\times$ ) and greater than $\times$ ) solar.
The model SEDs were reddened from 0.0 mag to 2.0 mag in 0.02 mag increments using the Cardelli.Clayton.&Mathis(1989) reddening law.
The model SEDs were reddened from 0.0 mag to 2.0 mag in 0.02 mag increments using the \citet{car89} reddening law.
Finally. the model SEDs were convolved with the Johnson and Kron-CousinsUBVA/ filter bandpasses and the broadband optical colours were determined.
Finally, the model SEDs were convolved with the Johnson and Kron-Cousins filter bandpasses and the broadband optical colours were determined.
The models span a range of ages from | Myr to20 Gyr.
The models span a range of ages from 1 Myr to20 Gyr.
From | Myr to | Gyr we used a step size of | Myr: from I.I to 20 Gyr. the step size was 100 Myr.
From 1 Myr to 1 Gyr we used a step size of 1 Myr; from 1.1 to 20 Gyr, the step size was 100 Myr.
Because the models will be compared to the integrated colours of resolved stellar populations. only the
Because the models will be compared to the integrated colours of resolved stellar populations, only the
the system is detached instead of semi-detached: however. in such a case 1t is impossible to find firm lower mass limits.
the system is detached instead of semi-detached; however, in such a case it is impossible to find firm lower mass limits.
To obtain a sample of detached solutions we modified our standard procedure in the following way: 1) set q and a: 11) adjust 7 so that the calculated velocity curve agrees with the observed one: iii) adjust the size of the primary to get Moo) = 6.0 mag: iv) adjust the size (and. if necessary. the temperature) of the secondary to get Mj,» = 6.3 mag (the latter two values are of course arbitrary. the only constraints being that the secondary is not much dimmer than the primary. and the combined bolometric magnitude of the system 1s 5.38 mag). v) check if the calculated light curve agrees with the observed one.
To obtain a sample of detached solutions we modified our standard procedure in the following way: i) set $q$ and $a$; ii) adjust $i$ so that the calculated velocity curve agrees with the observed one; iii) adjust the size of the primary to get $M_{bol1}$ = 6.0 mag; iv) adjust the size (and, if necessary, the temperature) of the secondary to get $M_{bol2}$ = 6.3 mag (the latter two values are of course arbitrary, the only constraints being that the secondary is not much dimmer than the primary, and the combined bolometric magnitude of the system is 5.38 mag), v) check if the calculated light curve agrees with the observed one.
If one assumes that the spectrum 1s generated by the secondary. then the inclination. consistent. with the velocity curve is much too low to reproduce the light curve (except for systems with g=0.75. but in those cases the masses of the components are smaller than 0.05 M..).
If one assumes that the spectrum is generated by the secondary, then the inclination consistent with the velocity curve is much too low to reproduce the light curve (except for systems with $q\gtrsim0.75$, but in those cases the masses of the components are smaller than 0.05 $M_\odot$ ).
The only reasonable solutions we were able to find for the detached configuration are based on the assumption that the primary generates the observed spectrum. while the secondary is mainly responsible for the observed ellipticity effect.
The only reasonable solutions we were able to find for the detached configuration are based on the assumption that the primary generates the observed spectrum, while the secondary is mainly responsible for the observed ellipticity effect.
The results shown in Table 12. seem to favor a configuration with small and rather discrepantV36.
The results shown in Table \ref{tab: V27} seem to favor a configuration with small and rather discrepant.
. The visible component must be the secondary: the alternative assumption enforces inclinations such that the calculated amplitude of the light curve is much lower than the observed one for all values of q.
The visible component must be the secondary; the alternative assumption enforces inclinations such that the calculated amplitude of the light curve is much lower than the observed one for all values of $q$.
However. assuming that the secondary fills its Roche lobe is untenable. as our standard analysis indicates that in such a case the caleulated amplitudes of the light curve are too high.
However, assuming that the secondary fills its Roche lobe is untenable, as our standard analysis indicates that in such a case the calculated amplitudes of the light curve are too high.
Thus. we are forcec to adopt a detached configuration.
Thus, we are forced to adopt a detached configuration.
Since V36 is located at the turnoff (only 0.3 mag below V17: see Fig. 2)).
Since V36 is located at the turnoff (only 0.3 mag below V17; see Fig. \ref{fig: n6397cmd}) ),
we assume that vin= and apply the following four-step procedure: 1) set ας M) adjust a to get 5=0.75M; in) adjust 7 so that the calculatedvelocity curve agrees with the observed one; iv) adjust the size of the secondary to get Mi,»=3.65 mag.
we assume that $m_2 = 0.75 \, M_\odot$ and apply the following four-step procedure: i) set $q$; ii) adjust $a$ to get $m_2 = 0.75 \, M_\odot$; iii) adjust $i$ so that the calculatedvelocity curve agrees with the observed one; iv) adjust the size of the secondary to get $M_{bol2} = 3.65$ mag.
The results shown in Table 13. indicate that V36 is similar to V17. with the primary’s mass possibly even higher than 1.5 M...
The results shown in Table \ref{tab: V36} indicate that V36 is similar to V17, with the primary's mass possibly even higher than 1.5 $M_\odot$.
10 out of 11 objects in our sample exhibit radial-velocity variations indicating their binary nature (the only exception ts V240 in c Cen).
10 out of 11 objects in our sample exhibit radial-velocity variations indicating their binary nature (the only exception is V240 in $\omega$ Cen).
In that sense. the sample was well chosen.
In that sense, the sample was well chosen.
Regarding the principal aim of the present survey we have been less successful - no clear-cut evidence for high-mass degenerate components was found.
Regarding the principal aim of the present survey we have been less successful - no clear-cut evidence for high-mass degenerate components was found.
However. while 8 systems proved to be more or less ordinary binaries. the remainig two (VI7 and V36 at the turnoff of NGC 6397) cearly deserve further scrutiny.
However, while 8 systems proved to be more or less ordinary binaries, the remainig two (V17 and V36 at the turnoff of NGC 6397) cearly deserve further scrutiny.
First. they are the only systems in which the brighter component is the less massive secondary.
First, they are the only systems in which the brighter component is the less massive secondary.
Second. the masses of their dim primaries may be significantly larger than | M. (in V36. even larger than 5 M. if the secondary is indeed a tumoff-mass star).
Second, the masses of their dim primaries may be significantly larger than 1 $M_\odot$ (in V36, even larger than 2 $M_\odot$ if the secondary is indeed a turnoff-mass star).
Third. they are weak X-ray sources.
Third, they are weak X-ray sources.
Based on Chandra observations. Bogdanovetal.(2010) classify them as active binaries (AB). Le. systems composed of main-sequence or subgiant stars whose weak X-ray emission originates from magnetic activity.
Based on Chandra observations, \cite{bog10} classify them as active binaries (AB), i.e. systems composed of main-sequence or subgiant stars whose weak X-ray emission originates from magnetic activity.
Our results rule out the possibility that they are composed of pristine stars which have not undergone any mass transfer episode. as in such a case the more massive component would also have to be the brighter one.
Our results rule out the possibility that they are composed of pristine stars which have not undergone any mass transfer episode, as in such a case the more massive component would also have to be the brighter one.
This is not to say Bogdanovetal.(2010) are wrong: short-period degenerate binaries in quiescence may easily “masquerade” as AB systems. since they are most likely synchronized and their visible components spin fast enough for the stellar dynamo to be highly efficient.
This is not to say \cite{bog10} are wrong: short-period degenerate binaries in quiescence may easily “masquerade” as AB systems, since they are most likely synchronized and their visible components spin fast enough for the stellar dynamo to be highly efficient.
According to the most optimistic interpretation. of our results. V17 and V36 may contain a neutron star (the dim primary of V36 may even be a black hole).
According to the most optimistic interpretation of our results, V17 and V36 may contain a neutron star (the dim primary of V36 may even be a black hole).
Such systems can indeed be expected. as population synthesis calculations indicate that the formation of a degenerate binary in which a neutron star or a black hole is accompanied by a main-sequence star or a subgiant is nothing unusual in. globular clusters.
Such systems can indeed be expected, as population synthesis calculations indicate that the formation of a degenerate binary in which a neutron star or a black hole is accompanied by a main-sequence star or a subgiant is nothing unusual in globular clusters.
Although the rate at which they form is not high («2.5 systems/Gyr). they should be transient all the time. and therefore more likely seen as a qLMXBs rather than as bright LMXBs (Ivanovaetal. 2008)..
Although the rate at which they form is not high $<2.5$ systems/Gyr), they should be transient all the time, and therefore more likely seen as a qLMXBs rather than as bright LMXBs \citep{iva08}. .
Moreover. NGC 6397 belongs to core-collapse clusters for which the formation rates of Ivanovaetal.(2008) are just lower limits.
Moreover, NGC 6397 belongs to core-collapse clusters for which the formation rates of \cite{iva08} are just lower limits.
Another
Another
There are (wo or four solutions {ο the equation (3.1.4)). which indicates that there are two solutions outside the ceuadroid caustic ancl four solutions inside the caustic.
There are two or four solutions to the equation \ref{eqRadialOne}) ), which indicates that there are two solutions outside the quadroid caustic and four solutions inside the caustic.
When the source is inside the caustic. (he images are all at ry221.
When the source is inside the caustic, the images are all at $r \approx 1$.
Thev are the four bright images that form around the critical curve r21. two outside the critical curve in the area of the “squeezed” and (wo inisde the critical curve in the area of the “huleed”.
They are the four bright images that form around the critical curve $r\approx 1$, two outside the critical curve in the area of the “squeezed" and two inisde the critical curve in the area of the “bulged".
For example. ¢=0 is inside the caustic. and the four images are on the real axis ancl the imaginary axis.
For example, $\zeta =0$ is inside the caustic, and the four images are on the real axis and the imaginary axis.
= DE σσ απ): os 4 "EN= l- ay)Yee:: 0, p, 42) Theradius ry is bigger than ry) because do«1 for a double scattering lens with d«1.
^2 = 1 - a_2); _1 = _2 = ^2 = 1 - a_2); _3 = 0, _4 = Theradius $r_{1,2}$ is bigger than $r_{3,4}$ because $\tilde a_2 < 1$ for a double scattering lens with $d < 1$.
Generally, the radii of the images are different.
Generally, the radii of the images are different.
Figure 3. shows a case: (6=100. d=1/3. and M,= Maé0-2.x10.
Figure \ref{fig_leqone} shows a case: $\ell=100$ , $d=1/3$, and $M_1=M_2$; $\zeta_1 = \zeta_2 = 2.\times 10^{-5}$.
The angle @ lor the radius r of each image is determined from eq.(3.1.43)).
The angle $\theta$ for the radius $r$ of each image is determined from \ref{eqImageAngle}) ).